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Constructions of Mutually Unbiased Bases

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Book cover Finite Fields and Applications (Fq 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2948))

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Abstract

Two orthonormal bases B and B′ of a d-dimensional complex inner-product space are called mutually unbiased if and only if |〈b|b′ 〉|2 = 1/d holds for all b ∈ B and b′ ∈ B′. The size of any set containing pairwise mutually unbiased bases of ℂd cannot exceed d + 1. If d is a power of a prime, then extremal sets containing d+1 mutually unbiased bases are known to exist. We give a simplified proof of this fact based on the estimation of exponential sums. We discuss conjectures and open problems concerning the maximal number of mutually unbiased bases for arbitrary dimensions.

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References

  1. Alltop, W.O.: Complex sequences with low periodic correlations. IEEE Transactions on Information Theory 26(3), 350–354 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aravind, P.K.: Solution to the king’s problem in prime power dimensions. Z. Naturforschung 58a, 2212 (2003)

    Google Scholar 

  3. Bandyopadhyay, S., Boykin, P.O., Roychowdhury, V., Vatan, F.: A new proof of the existence of mutually unbiased bases. Algorithmica 34, 512–528 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bechmann-Pasquinucci, H., Tittel, W.: Quantum cryptography using larger alphabets. Phys. Rev. A 61(6), 062308 (2000)

    Google Scholar 

  5. Bennett, C.H., Brassard, G.: Quantum cryptography: Public key distribution and coin tossing. In: Proceedings of the IEEE Intl. Conf. Computers, Systems, and Signal Processing, pp. 175–179. IEEE, Los Alamitos (1984)

    Google Scholar 

  6. Beth, T., Jungnickel, D., Lenz, H.: Design Theory, 2nd edn., vol. 2. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  7. Bollobás, B.: Random Graphs. Academic Press, London (1985)

    MATH  Google Scholar 

  8. Carlet, C.: One-weight Z4-linear codes. In: Buchmann, J., Høholdt, T., Stichtenoth, H., Tapia-Recillas, H. (eds.) Coding Theory, Cryprography and Related Areas, pp. 57–72. Springer, Heidelberg (2000)

    Google Scholar 

  9. Chaturvedi, S.: Aspects of mutually unbiased bases in odd-prime-power dimensions. Phys. Rev. A 65, 044301 (2002)

    Google Scholar 

  10. Chowla, S., Erdös, P., Strauss, E.G.: On the maximal number of pairwise orthogonal latin squares of given order. Canadian J. Math. 12, 204–208 (1960)

    Article  MATH  Google Scholar 

  11. Delsarte, P., Goethals, J.M., Seidel, J.J.: Bounds for systems of lines, and Jacobi polynomials. Philips Res. Repts., 91–105 (1975)

    Google Scholar 

  12. Englert, B.-G., Aharonov, Y.: The mean king’s problem: Prime degrees of freedom. Phys. Letters 284, 1–5 (2001)

    Article  MathSciNet  Google Scholar 

  13. Hoggar, S.G.: t-designs in projective spaces. Europ. J. Combin. 3, 233–254 (1982)

    MATH  MathSciNet  Google Scholar 

  14. Ivanović, I.D.: Geometrical description of quantal state determination. J. Phys. A 14, 3241–3245 (1981)

    Article  MathSciNet  Google Scholar 

  15. Kabatiansky, G.A., Levenshtein, V.I.: Bounds for packings on a sphere and in space. Problems of Information Transmission 14(1), 1–17 (1978)

    Google Scholar 

  16. Laywine, C.F., Mullen, G.L.: Discrete Mathematics Using Latin Squares. John Wiley, New York (1998)

    MATH  Google Scholar 

  17. Lidl, R., Niederreiter, H.: Introduction to finite fields and their applications, 2nd edn. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  18. Schwinger, J.: Unitary operator bases. Proc. Nat. Acad. Sci. U.S.A. 46, 570–579 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  19. Vaidman, L., Aharonov, Y., Albert, D.Z.: How to ascertain the values of σ x , σ y , and σ z . Phys. Rev. Lett. 58, 1385–1387 (1987)

    Article  MathSciNet  Google Scholar 

  20. Wan, Z.-X.: Quaternary Codes. World-Scientific, Singapore (1997)

    MATH  Google Scholar 

  21. Wilson, R.M.: Concerning the number of mutually orthogonal Latin squares. Discr. Math. 9, 181–198 (1974)

    Article  MATH  Google Scholar 

  22. Wootters, W.K., Fields, B.D.: Optimal state-determination by mutually unbiased measurements. Ann. Physics 191, 363–381 (1989)

    Article  MathSciNet  Google Scholar 

  23. Yang, K., Helleseth, T., Kumar, P.V., Shanbhag, A.G.: On the weight hierarchy of Kerdock codes over Z4. IEEE Transactions on Information Theory 42(5), 1587–1593 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  24. Zauner, G.: Quantendesigns – Grundzüge einer nichtkommutativen Designtheorie. PhD thesis, Universität Wien (1999)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, Texas A&M University, College Station, TX, 77843-3112, USA

    Andreas Klappenecker

  2. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1

    Martin Rötteler

Authors
  1. Andreas Klappenecker
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  2. Martin Rötteler
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Editor information

Editors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, University Park, 16802, PA, USA

    Gary L. Mullen

  2. University Paul Sabatier, AAECC/IRIT, 118 route de Narbonne, 31300, Toulouse cedex, France

    Alain Poli

  3. Sabancı University - FENS, Orhanli - Tuzla, 34956, Istanbul, Turkey

    Henning Stichtenoth

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© 2004 Springer-Verlag Berlin Heidelberg

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Klappenecker, A., Rötteler, M. (2004). Constructions of Mutually Unbiased Bases. In: Mullen, G.L., Poli, A., Stichtenoth, H. (eds) Finite Fields and Applications. Fq 2003. Lecture Notes in Computer Science, vol 2948. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24633-6_10

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  • DOI: https://doi.org/10.1007/978-3-540-24633-6_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-21324-6

  • Online ISBN: 978-3-540-24633-6

  • eBook Packages: Springer Book Archive

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